\documentclass[9pt]{article}
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\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  1.0 & -8.00 x_{1} & +  1.00 x_{2} & +  1.00 x_{3} & +  7.00 x_{4} & -7.00 x_{5}\\
 x_{7}   &  0.0 & -7.00 x_{1} & +  9.00 x_{2} & -6.00 x_{3} & -9.00 x_{4} & +  1.00 x_{5}\\
 x_{8}   &  1.0 & -7.00 x_{1} & +  6.00 x_{2} & + 10.00 x_{3} & +  9.00 x_{4} & +  1.00 x_{5}\\
 x_{9}   &  1.0 & -6.00 x_{1} & -4.00 x_{2} & -8.00 x_{3} & +  3.00 x_{4} & +  6.00 x_{5}\\
 x_{10}   &  1.0 & -10.00 x_{1} & +  1.00 x_{2} & +  2.00 x_{3} & -9.00 x_{4} & -4.00 x_{5}\\
 x_{11}   &  -3.0 & -1.00 x_{1} & +  6.00 x_{2} & -5.00 x_{3} & +  2.00 x_{4} & +  1.00 x_{5}\\
 x_{12}   &  2.0 & -1.00 x_{1} & + 10.00 x_{2} & -4.00 x_{3} & +  1.00 x_{4} & +  6.00 x_{5}\\
 x_{13}   &  3.0 & -1.00 x_{1} & -5.00 x_{2} & +  8.00 x_{3} & -5.00 x_{4} & +  7.00 x_{5}\\
\hline
z    &  0.0 & -3.00 x_{1} & +  5.00 x_{2} & -3.00 x_{3} & -4.00 x_{4} & -3.00 x_{5}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{5}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  8.00 y_{6} & +  7.00 y_{7} & +  7.00 y_{8} & +  6.00 y_{9} & + 10.00 y_{10} & +  1.00 y_{11} & +  1.00 y_{12} & +  1.00 y_{13}\\
 y_{2}   &  1.0 & -1.00 y_{6} & -9.00 y_{7} & -6.00 y_{8} & +  4.00 y_{9} & -1.00 y_{10} & -6.00 y_{11} & -10.00 y_{12} & +  5.00 y_{13}\\
 y_{3}   &  1.0 & -1.00 y_{6} & +  6.00 y_{7} & -10.00 y_{8} & +  8.00 y_{9} & -2.00 y_{10} & +  5.00 y_{11} & +  4.00 y_{12} & -8.00 y_{13}\\
 y_{4}   &  1.0 & -7.00 y_{6} & +  9.00 y_{7} & -9.00 y_{8} & -3.00 y_{9} & +  9.00 y_{10} & -2.00 y_{11} & -1.00 y_{12} & +  5.00 y_{13}\\
 y_{5}   &  1.0 & +  7.00 y_{6} & -1.00 y_{7} & -1.00 y_{8} & -6.00 y_{9} & +  4.00 y_{10} & -1.00 y_{11} & -6.00 y_{12} & -7.00 y_{13}\\
\hline
z    &  -0 & -1.00 y_{6} &   & -1.00 y_{8} & -1.00 y_{9} & -1.00 y_{10} & +  3.00 y_{11} & -2.00 y_{12} & -3.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 3 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  2.0 & + 15.00 y_{6} & +  6.00 y_{7} & +  6.00 y_{8} & -1.00 y_{5} & + 14.00 y_{10} &   & -5.00 y_{12} & -6.00 y_{13}\\
 y_{11}   &  0.25 & +  0.55 y_{6} & -1.45 y_{7} & -1.00 y_{8} & -0.10 y_{5} & +  0.25 y_{10} & -0.15 y_{2} & -2.10 y_{12} & +  0.05 y_{13}\\
 y_{3}   &  3.25 & + 10.35 y_{6} & -0.65 y_{7} & -15.00 y_{8} & -1.70 y_{5} & +  4.25 y_{10} & -0.55 y_{2} & -11.70 y_{12} & -17.15 y_{13}\\
 y_{4}   &  0.125 & -11.32 y_{6} & + 11.68 y_{7} & -7.00 y_{8} & +  0.65 y_{5} & +  6.63 y_{10} & +  0.22 y_{2} & +  5.15 y_{12} & +  8.42 y_{13}\\
 y_{9}   &  0.125 & +  1.07 y_{6} & +  0.07 y_{7} &   & -0.15 y_{5} & +  0.62 y_{10} & +  0.02 y_{2} & -0.65 y_{12} & -1.17 y_{13}\\
\hline
z    &  0.625 & -0.43 y_{6} & -4.42 y_{7} & -4.00 y_{8} & -0.15 y_{5} & -0.88 y_{10} & -0.47 y_{2} & -7.65 y_{12} & -1.67 y_{13}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.425 & -15.00 x_{1} & -0.55 x_{11} & -10.35 x_{3} & + 11.32 x_{4} & -1.07 x_{9}\\
 x_{7}   &  4.425 & -6.00 x_{1} & +  1.45 x_{11} & +  0.65 x_{3} & -11.68 x_{4} & -0.07 x_{9}\\
 x_{8}   &  4.0 & -6.00 x_{1} & +  1.00 x_{11} & + 15.00 x_{3} & +  7.00 x_{4} &   \\
 x_{5}   &  0.15 & +  1.00 x_{1} & +  0.10 x_{11} & +  1.70 x_{3} & -0.65 x_{4} & +  0.15 x_{9}\\
 x_{10}   &  0.875 & -14.00 x_{1} & -0.25 x_{11} & -4.25 x_{3} & -6.63 x_{4} & -0.62 x_{9}\\
 x_{2}   &  0.475  &   & +  0.15 x_{11} & +  0.55 x_{3} & -0.22 x_{4} & -0.02 x_{9}\\
 x_{12}   &  7.65 & +  5.00 x_{1} & +  2.10 x_{11} & + 11.70 x_{3} & -5.15 x_{4} & +  0.65 x_{9}\\
 x_{13}   &  1.675 & +  6.00 x_{1} & -0.05 x_{11} & + 17.15 x_{3} & -8.42 x_{4} & +  1.17 x_{9}\\
\hline
z    &  -0.625 & -2.00 x_{1} & -0.25 x_{11} & -3.25 x_{3} & -0.13 x_{4} & -0.12 x_{9}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.425 & -15.00 x_{1} & -0.55 x_{11} & -10.35 x_{3} & + 11.32 x_{4} & -1.07 x_{9}\\
 x_{7}   &  4.425 & -6.00 x_{1} & +  1.45 x_{11} & +  0.65 x_{3} & -11.68 x_{4} & -0.07 x_{9}\\
 x_{8}   &  4.0 & -6.00 x_{1} & +  1.00 x_{11} & + 15.00 x_{3} & +  7.00 x_{4} &   \\
 x_{5}   &  0.15 & +  1.00 x_{1} & +  0.10 x_{11} & +  1.70 x_{3} & -0.65 x_{4} & +  0.15 x_{9}\\
 x_{10}   &  0.875 & -14.00 x_{1} & -0.25 x_{11} & -4.25 x_{3} & -6.63 x_{4} & -0.62 x_{9}\\
 x_{2}   &  0.475  &   & +  0.15 x_{11} & +  0.55 x_{3} & -0.22 x_{4} & -0.02 x_{9}\\
 x_{12}   &  7.65 & +  5.00 x_{1} & +  2.10 x_{11} & + 11.70 x_{3} & -5.15 x_{4} & +  0.65 x_{9}\\
 x_{13}   &  1.675 & +  6.00 x_{1} & -0.05 x_{11} & + 17.15 x_{3} & -8.42 x_{4} & +  1.17 x_{9}\\
\hline
z    &  1.925 & -6.00 x_{1} & +  0.45 x_{11} & -5.35 x_{3} & -3.18 x_{4} & -0.57 x_{9}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.425 & -15.00 x_{1} & -0.55 x_{11} & -10.35 x_{3} & + 11.32 x_{4} & -1.07 x_{9}\\
 x_{7}   &  4.425 & -6.00 x_{1} & +  1.45 x_{11} & +  0.65 x_{3} & -11.68 x_{4} & -0.07 x_{9}\\
 x_{8}   &  4.0 & -6.00 x_{1} & +  1.00 x_{11} & + 15.00 x_{3} & +  7.00 x_{4} &   \\
 x_{5}   &  0.15 & +  1.00 x_{1} & +  0.10 x_{11} & +  1.70 x_{3} & -0.65 x_{4} & +  0.15 x_{9}\\
 x_{10}   &  0.875 & -14.00 x_{1} & -0.25 x_{11} & -4.25 x_{3} & -6.63 x_{4} & -0.62 x_{9}\\
 x_{2}   &  0.475  &   & +  0.15 x_{11} & +  0.55 x_{3} & -0.22 x_{4} & -0.02 x_{9}\\
 x_{12}   &  7.65 & +  5.00 x_{1} & +  2.10 x_{11} & + 11.70 x_{3} & -5.15 x_{4} & +  0.65 x_{9}\\
 x_{13}   &  1.675 & +  6.00 x_{1} & -0.05 x_{11} & + 17.15 x_{3} & -8.42 x_{4} & +  1.17 x_{9}\\
\hline
z    &  1.925 & -6.00 x_{1} & +  0.45 x_{11} & -5.35 x_{3} & -3.18 x_{4} & -0.57 x_{9}\\
\end{array}\]


 $ x_{11} $ enters and $ x_{6} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{11}   &  0.772727272727 & -27.27 x_{1} & -1.82 x_{6} & -18.82 x_{3} & + 20.59 x_{4} & -1.95 x_{9}\\
 x_{7}   &  5.54545454545 & -45.55 x_{1} & -2.64 x_{6} & -26.64 x_{3} & + 18.18 x_{4} & -2.91 x_{9}\\
 x_{8}   &  4.77272727273 & -33.27 x_{1} & -1.82 x_{6} & -3.82 x_{3} & + 27.59 x_{4} & -1.95 x_{9}\\
 x_{5}   &  0.227272727273 & -1.73 x_{1} & -0.18 x_{6} & -0.18 x_{3} & +  1.41 x_{4} & -0.05 x_{9}\\
 x_{10}   &  0.681818181818 & -7.18 x_{1} & +  0.45 x_{6} & +  0.45 x_{3} & -11.77 x_{4} & -0.14 x_{9}\\
 x_{2}   &  0.590909090909 & -4.09 x_{1} & -0.27 x_{6} & -2.27 x_{3} & +  2.86 x_{4} & -0.32 x_{9}\\
 x_{12}   &  9.27272727273 & -52.27 x_{1} & -3.82 x_{6} & -27.82 x_{3} & + 38.09 x_{4} & -3.45 x_{9}\\
 x_{13}   &  1.63636363636 & +  7.36 x_{1} & +  0.09 x_{6} & + 18.09 x_{3} & -9.45 x_{4} & +  1.27 x_{9}\\
\hline
z    &  2.27272727273 & -18.27 x_{1} & -0.82 x_{6} & -13.82 x_{3} & +  6.09 x_{4} & -1.45 x_{9}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{11}   &  1.96525096525 & -39.83 x_{1} & -1.02 x_{6} & -18.02 x_{3} & -1.75 x_{10} & -2.19 x_{9}\\
 x_{7}   &  6.59845559846 & -56.64 x_{1} & -1.93 x_{6} & -25.93 x_{3} & -1.54 x_{10} & -3.12 x_{9}\\
 x_{8}   &  6.37065637066 & -50.10 x_{1} & -0.75 x_{6} & -2.75 x_{3} & -2.34 x_{10} & -2.27 x_{9}\\
 x_{5}   &  0.30888030888 & -2.59 x_{1} & -0.13 x_{6} & -0.13 x_{3} & -0.12 x_{10} & -0.06 x_{9}\\
 x_{4}   &  0.0579150579151 & -0.61 x_{1} & +  0.04 x_{6} & +  0.04 x_{3} & -0.08 x_{10} & -0.01 x_{9}\\
 x_{2}   &  0.756756756757 & -5.84 x_{1} & -0.16 x_{6} & -2.16 x_{3} & -0.24 x_{10} & -0.35 x_{9}\\
 x_{12}   &  11.4787644788 & -75.51 x_{1} & -2.35 x_{6} & -26.35 x_{3} & -3.24 x_{10} & -3.90 x_{9}\\
 x_{13}   &  1.0888030888 & + 13.13 x_{1} & -0.27 x_{6} & + 17.73 x_{3} & +  0.80 x_{10} & +  1.38 x_{9}\\
\hline
z    &  2.62548262548 & -21.99 x_{1} & -0.58 x_{6} & -13.58 x_{3} & -0.52 x_{10} & -1.53 x_{9}\\
\end{array}\]
Final Dictionary
Solution:  2.62548262548
Num Pivots:  3
\end{document}
